\newproblem{lay:1_2_2}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.2.2}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Determine which of the following matrices are in reduced echelon form and which others are only in echelon form.
	\begin{enumerate}[a]
		\item $\begin{pmatrix} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$
		\item $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}$
		\item $\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$
		\item $\begin{pmatrix} 0 & 1 & 1 & 1 & 1\\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$
	\end{enumerate}
}{
   % Solution
	Let's remind the conditions to be in reduced echelon form.
	\begin{enumerate}
		\item Within each row, the first element different from zero (called the leading entry) is in a column to the right of the leading entry of the previous row.
		\item Within each column, all values below a leading entry are zero.
		\item All rows without a leading entry (i.e., they only have zeros) are below all the rows in which at least one element is not zero.
		\item The leading entry of each row is 1.
		\item The leading entry is the only 1 in its column.
	\end{enumerate}
	Those matrices meeting only 1-3 are said to be in echelon form. Looking at the matrices of the exercise.
	\begin{enumerate}[a]
		\item It is in reduced echelon.
		\item It is in echelon form because there is a leading entry in the second column but it is not 1.
		\item It is not in echelon form nor in reduced echelon form because the first row is full of zeroes, and there are rows with leading entries below.
		\item It is in echelon form because the leading entries in each row are not the only non-zero values in their columns.
	\end{enumerate}
}
\useproblem{lay:1_2_2}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
